Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 12.0s

Alternatives: 20

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around inf

   \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

   2. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

   3. unpow2 N/A

      \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

   4. lower-hypot.f64 N/A

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   5. lower-sin.f64 N/A

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

   6. lower-sin.f64 99.7

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

5. Simplified 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
6. Add Preprocessing

Alternative 2: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.99975:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.5:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;t\_4 \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{t\_3}}\right)\\ \mathbf{elif}\;t\_4 \leq 0.9995:\\ \;\;\;\;t\_2 \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot kx (sin ky)))))
        (t_2 (/ (sin ky) (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -0.99975)
     t_1
     (if (<= t_4 -0.5)
       (* t_2 (fma th (* -0.16666666666666666 (* th th)) th))
       (if (<= t_4 1e-9)
         (* (sin th) (* (sin ky) (sqrt (/ 1.0 t_3))))
         (if (<= t_4 0.9995)
           (*
            t_2
            (fma
             th
             (*
              (* th th)
              (fma 0.008333333333333333 (* th th) -0.16666666666666666))
             th))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	double t_2 = sin(ky) / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.99975) {
		tmp = t_1;
	} else if (t_4 <= -0.5) {
		tmp = t_2 * fma(th, (-0.16666666666666666 * (th * th)), th);
	} else if (t_4 <= 1e-9) {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / t_3)));
	} else if (t_4 <= 0.9995) {
		tmp = t_2 * fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))))
	t_2 = Float64(sin(ky) / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.99975)
		tmp = t_1;
	elseif (t_4 <= -0.5)
		tmp = Float64(t_2 * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th));
	elseif (t_4 <= 1e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / t_3))));
	elseif (t_4 <= 0.9995)
		tmp = Float64(t_2 * fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99975], t$95$1, If[LessEqual[t$95$4, -0.5], N[(t$95$2 * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9995], N[(t$95$2 * N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.999750000000000028 or 0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 75.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 74.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 74.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 99.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   if -0.999750000000000028 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.5

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 67.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 67.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if -0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000006e-9

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}}\right) \cdot \sin th \]

      9. lower-sin.f64 99.4

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}}\right) \cdot \sin th \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       2. lower-pow.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       3. lower-sin.f64 95.7

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2}}}\right) \cdot \sin th \]

   11. Simplified 95.7%

       \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

   if 1.00000000000000006e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99950000000000006

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right) \]

      8. sub-neg N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right) \]

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right) \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right) \]

      12. lower-*.f64 48.0

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right) \]

   8. Simplified 48.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99975:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.5:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2}}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -0.99975:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{t\_3}}\right)\\ \mathbf{elif}\;t\_4 \leq 0.9995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot kx (sin ky)))))
        (t_2
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (fma th (* -0.16666666666666666 (* th th)) th)))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -0.99975)
     t_1
     (if (<= t_4 -0.5)
       t_2
       (if (<= t_4 1e-9)
         (* (sin th) (* (sin ky) (sqrt (/ 1.0 t_3))))
         (if (<= t_4 0.9995) t_2 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	double t_2 = (sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, (-0.16666666666666666 * (th * th)), th);
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -0.99975) {
		tmp = t_1;
	} else if (t_4 <= -0.5) {
		tmp = t_2;
	} else if (t_4 <= 1e-9) {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / t_3)));
	} else if (t_4 <= 0.9995) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))))
	t_2 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -0.99975)
		tmp = t_1;
	elseif (t_4 <= -0.5)
		tmp = t_2;
	elseif (t_4 <= 1e-9)
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / t_3))));
	elseif (t_4 <= 0.9995)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99975], t$95$1, If[LessEqual[t$95$4, -0.5], t$95$2, If[LessEqual[t$95$4, 1e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9995], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.999750000000000028 or 0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 75.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 74.2

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 74.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 99.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 99.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   if -0.999750000000000028 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.5 or 1.00000000000000006e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99950000000000006

   1. Initial program 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.3

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.3%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 55.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 55.7%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   if -0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.00000000000000006e-9

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}}\right) \cdot \sin th \]

      9. lower-sin.f64 99.4

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}}\right) \cdot \sin th \]

   8. Simplified 99.4%

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       2. lower-pow.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       3. lower-sin.f64 95.7

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2}}}\right) \cdot \sin th \]

   11. Simplified 95.7%

       \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99975:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.5:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-9}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2}}}\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9995:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{t\_2 + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}\\ \mathbf{elif}\;t\_3 \leq 0.9995:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin th) (/ (sin ky) (hypot kx (sin ky)))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.2)
     t_1
     (if (<= t_3 0.001)
       (*
        (sin th)
        (/
         (fma ky (* -0.16666666666666666 (* ky ky)) ky)
         (sqrt (+ t_2 (* ky (fma ky (* (* ky ky) -0.3333333333333333) ky))))))
       (if (<= t_3 0.9995) (* (* (sin ky) (sin th)) (/ 1.0 (sin ky))) t_1)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_1;
	} else if (t_3 <= 0.001) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / sqrt((t_2 + (ky * fma(ky, ((ky * ky) * -0.3333333333333333), ky)))));
	} else if (t_3 <= 0.9995) {
		tmp = (sin(ky) * sin(th)) * (1.0 / sin(ky));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.2)
		tmp = t_1;
	elseif (t_3 <= 0.001)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / sqrt(Float64(t_2 + Float64(ky * fma(ky, Float64(Float64(ky * ky) * -0.3333333333333333), ky))))));
	elseif (t_3 <= 0.9995)
		tmp = Float64(Float64(sin(ky) * sin(th)) * Float64(1.0 / sin(ky)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.2], t$95$1, If[LessEqual[t$95$3, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[(ky * N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9995], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.99950000000000006 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 80.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 59.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 59.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 78.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 78.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}} \cdot \sin th \]

      2. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \left(ky \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}}} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \left(ky \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \left(ky \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin th \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \color{blue}{\left(ky \cdot \left(\frac{-1}{3} \cdot {ky}^{2}\right) + ky \cdot 1\right)}}} \cdot \sin th \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \left(ky \cdot \left(\frac{-1}{3} \cdot {ky}^{2}\right) + \color{blue}{ky}\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \color{blue}{\mathsf{fma}\left(ky, \frac{-1}{3} \cdot {ky}^{2}, ky\right)}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{3}}, ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{3}}, ky\right)}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      11. lower-*.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   5. Simplified 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      8. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.99950000000000006

   1. Initial program 99.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin th} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\sin th \cdot \color{blue}{\sin ky}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      12. lower-sin.f64 99.1

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

   8. Simplified 99.1%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   9. Taylor expanded in kx around 0

      \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\sin ky}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\sin ky}} \]

       2. lower-sin.f64 22.2

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \frac{1}{\color{blue}{\sin ky}} \]

   11. Simplified 22.2%

       \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\frac{1}{\sin ky}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.9995:\\ \;\;\;\;\left(\sin ky \cdot \sin th\right) \cdot \frac{1}{\sin ky}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{t\_3}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{t\_1}{\sqrt{t\_2 + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{t\_1}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (fma ky (* -0.16666666666666666 (* ky ky)) ky))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.2)
     (*
      th
      (*
       (sqrt (/ 1.0 t_3))
       (* (sin ky) (fma -0.16666666666666666 (* th th) 1.0))))
     (if (<= t_4 0.001)
       (*
        (sin th)
        (/
         t_1
         (sqrt (+ t_2 (* ky (fma ky (* (* ky ky) -0.3333333333333333) ky))))))
       (if (<= t_4 2.0) (sin th) (* (sin th) (/ t_1 (hypot kx (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma(ky, (-0.16666666666666666 * (ky * ky)), ky);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.2) {
		tmp = th * (sqrt((1.0 / t_3)) * (sin(ky) * fma(-0.16666666666666666, (th * th), 1.0)));
	} else if (t_4 <= 0.001) {
		tmp = sin(th) * (t_1 / sqrt((t_2 + (ky * fma(ky, ((ky * ky) * -0.3333333333333333), ky)))));
	} else if (t_4 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (t_1 / hypot(kx, sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky)
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.2)
		tmp = Float64(th * Float64(sqrt(Float64(1.0 / t_3)) * Float64(sin(ky) * fma(-0.16666666666666666, Float64(th * th), 1.0))));
	elseif (t_4 <= 0.001)
		tmp = Float64(sin(th) * Float64(t_1 / sqrt(Float64(t_2 + Float64(ky * fma(ky, Float64(Float64(ky * ky) * -0.3333333333333333), ky))))));
	elseif (t_4 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(t_1 / hypot(kx, sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.2], N[(th * N[(N[Sqrt[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.001], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[(ky * N[(ky * N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] + ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(t$95$1 / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001

   1. Initial program 87.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) + \sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} + \frac{-1}{6} \cdot \left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} + \color{blue}{\left(\left({th}^{2} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) \cdot \frac{-1}{6}}\right) \]

      3. associate-*l* N/A

         \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} + \color{blue}{\left({th}^{2} \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)} \cdot \frac{-1}{6}\right) \]

      4. associate-*r* N/A

         \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} + \color{blue}{{th}^{2} \cdot \left(\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) \cdot \frac{-1}{6}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} + {th}^{2} \cdot \color{blue}{\left(\frac{-1}{6} \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)\right)}\right) \]

   5. Simplified 48.5%

      \[\leadsto \color{blue}{th \cdot \left(\sqrt{\frac{1}{{\sin ky}^{2} + {\sin kx}^{2}}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right)} \]

   6. Taylor expanded in kx around 0

      \[\leadsto th \cdot \left(\sqrt{\color{blue}{\frac{1}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, th \cdot th, 1\right) \cdot \sin ky\right)\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto th \cdot \left(\sqrt{\color{blue}{\frac{1}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, th \cdot th, 1\right) \cdot \sin ky\right)\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto th \cdot \left(\sqrt{\frac{1}{\color{blue}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, th \cdot th, 1\right) \cdot \sin ky\right)\right) \]

      3. lower-sin.f64 31.2

         \[\leadsto th \cdot \left(\sqrt{\frac{1}{{\color{blue}{\sin ky}}^{2}}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right) \]

   8. Simplified 31.2%

      \[\leadsto th \cdot \left(\sqrt{\color{blue}{\frac{1}{{\sin ky}^{2}}}} \cdot \left(\mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right) \cdot \sin ky\right)\right) \]

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{ky}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{\left(ky \cdot ky\right)} \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)}} \cdot \sin th \]

      2. associate-*l* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \left(ky \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}}} \cdot \sin th \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \left(ky \cdot \left(1 + \frac{-1}{3} \cdot {ky}^{2}\right)\right)}}} \cdot \sin th \]

      4. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \left(ky \cdot \color{blue}{\left(\frac{-1}{3} \cdot {ky}^{2} + 1\right)}\right)}} \cdot \sin th \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \color{blue}{\left(ky \cdot \left(\frac{-1}{3} \cdot {ky}^{2}\right) + ky \cdot 1\right)}}} \cdot \sin th \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \left(ky \cdot \left(\frac{-1}{3} \cdot {ky}^{2}\right) + \color{blue}{ky}\right)}} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \color{blue}{\mathsf{fma}\left(ky, \frac{-1}{3} \cdot {ky}^{2}, ky\right)}}} \cdot \sin th \]

      8. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{3}}, ky\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{3}}, ky\right)}} \cdot \sin th \]

      10. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      11. lower-*.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   5. Simplified 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot \frac{-1}{3}, ky\right)}} \cdot \sin th \]

      8. lower-*.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}} \cdot \sin th \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 65.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 65.4%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 2.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 100.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       3. *-rgt-identity N/A

          \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       7. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       8. lower-*.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.2:\\ \;\;\;\;th \cdot \left(\sqrt{\frac{1}{{\sin ky}^{2}}} \cdot \left(\sin ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot th, 1\right)\right)\right)\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\sqrt{{\sin kx}^{2} + ky \cdot \mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.3333333333333333, ky\right)}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/ (sin ky) (hypot (sin ky) kx)))
     (if (<= t_1 0.15)
       (* (sin th) (/ (sin ky) (sin kx)))
       (if (<= t_1 2.0)
         (sin th)
         (*
          (sin th)
          (/
           (fma ky (* -0.16666666666666666 (* ky ky)) ky)
           (hypot kx (sin ky)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (t_1 <= 0.15) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / hypot(kx, sin(ky)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (t_1 <= 0.15)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / hypot(kx, sin(ky))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 79.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 76.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 76.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 42.0

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 42.0%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       4. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       5. lower-sin.f64 52.4

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   11. Simplified 52.4%

       \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 48.0

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 48.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.149999999999999994 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 66.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 66.1%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 2.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 2.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 100.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 100.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       3. *-rgt-identity N/A

          \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       7. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

       8. lower-*.f64 100.0

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\mathsf{hypot}\left(kx, \sin ky\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (*
      (fma th (* -0.16666666666666666 (* th th)) th)
      (/ (sin ky) (hypot (sin ky) kx)))
     (if (<= t_1 0.15) (* (sin th) (/ (sin ky) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th) * (sin(ky) / hypot(sin(ky), kx));
	} else if (t_1 <= 0.15) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th) * Float64(sin(ky) / hypot(sin(ky), kx)));
	elseif (t_1 <= 0.15)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 79.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 76.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 76.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 42.0

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 42.0%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {kx}^{2}}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       2. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {kx}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       3. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{kx \cdot kx}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       4. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       5. lower-sin.f64 52.4

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   11. Simplified 52.4%

       \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, kx\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 48.0

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 48.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.149999999999999994 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.98:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \cdot \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.15)
   (* (sin th) (/ (sin ky) (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.15) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.15d0) then
        tmp = sin(th) * (sin(ky) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.15) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.15:
		tmp = math.sin(th) * (math.sin(ky) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.15)
		tmp = Float64(sin(th) * Float64(sin(ky) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.15)
		tmp = sin(th) * (sin(ky) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.15], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.149999999999999994

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-sin.f64 36.9

         \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   5. Simplified 36.9%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.149999999999999994 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 62.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.15:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.001)
   (* (sin th) (/ ky (sin kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.001) {
		tmp = sin(th) * (ky / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.001d0) then
        tmp = sin(th) * (ky / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.001) {
		tmp = Math.sin(th) * (ky / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.001:
		tmp = math.sin(th) * (ky / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = Float64(sin(th) * Float64(ky / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.001)
		tmp = sin(th) * (ky / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1e-3

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.7

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

      2. lower-sin.f64 35.2

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   8. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 1e-3 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 62.1

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 62.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.001:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* (sin ky) (/ (sin th) kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = sin(ky) * (sin(th) / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = sin(ky) * (sin(th) / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = Math.sin(ky) * (Math.sin(th) / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = math.sin(ky) * (math.sin(th) / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(sin(ky) * Float64(sin(th) / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = sin(ky) * (sin(th) / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{kx + \frac{1}{2} \cdot \frac{{ky}^{2}}{kx}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{ky}^{2}}{kx} + kx}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2}}{kx} \cdot \frac{1}{2}} + kx} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2} \cdot \frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{ky}^{2} \cdot \frac{\frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{kx} + kx} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{kx}\right)} + kx} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      10. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{\frac{1}{2} \cdot 1}{kx}}, kx\right)} \cdot \sin th \]

      11. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \frac{\color{blue}{\frac{1}{2}}}{kx}, kx\right)} \cdot \sin th \]

      12. lower-/.f64 25.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{0.5}{kx}}, kx\right)} \cdot \sin th \]

   8. Simplified 25.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{kx}} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

       2. lower-*.f64 N/A

          \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

       3. lower-sin.f64 N/A

          \[\leadsto \color{blue}{\sin ky} \cdot \frac{\sin th}{kx} \]

       4. lower-/.f64 N/A

          \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{kx}} \]

       5. lower-sin.f64 25.2

          \[\leadsto \sin ky \cdot \frac{\color{blue}{\sin th}}{kx} \]

   11. Simplified 25.2%

       \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{kx}} \]

   if 5.0000000000000001e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (*
    (sin th)
    (/
     (fma ky (* -0.16666666666666666 (* ky ky)) ky)
     (fma (* ky ky) (/ 0.5 kx) kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = sin(th) * (fma(ky, (-0.16666666666666666 * (ky * ky)), ky) / fma((ky * ky), (0.5 / kx), kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(sin(th) * Float64(fma(ky, Float64(-0.16666666666666666 * Float64(ky * ky)), ky) / fma(Float64(ky * ky), Float64(0.5 / kx), kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[Sin[th], $MachinePrecision] * N[(N[(ky * N[(-0.16666666666666666 * N[(ky * ky), $MachinePrecision]), $MachinePrecision] + ky), $MachinePrecision] / N[(N[(ky * ky), $MachinePrecision] * N[(0.5 / kx), $MachinePrecision] + kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{kx + \frac{1}{2} \cdot \frac{{ky}^{2}}{kx}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{ky}^{2}}{kx} + kx}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2}}{kx} \cdot \frac{1}{2}} + kx} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2} \cdot \frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{ky}^{2} \cdot \frac{\frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{kx} + kx} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{kx}\right)} + kx} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      10. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{\frac{1}{2} \cdot 1}{kx}}, kx\right)} \cdot \sin th \]

      11. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \frac{\color{blue}{\frac{1}{2}}}{kx}, kx\right)} \cdot \sin th \]

      12. lower-/.f64 25.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{0.5}{kx}}, kx\right)} \cdot \sin th \]

   8. Simplified 25.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right)}}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + ky \cdot 1}}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       3. *-rgt-identity N/A

          \[\leadsto \frac{ky \cdot \left(\frac{-1}{6} \cdot {ky}^{2}\right) + \color{blue}{ky}}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \frac{-1}{6} \cdot {ky}^{2}, ky\right)}}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{{ky}^{2} \cdot \frac{-1}{6}}, ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       7. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot \frac{-1}{6}, ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{\frac{1}{2}}{kx}, kx\right)} \cdot \sin th \]

       8. lower-*.f64 24.3

          \[\leadsto \frac{\mathsf{fma}\left(ky, \color{blue}{\left(ky \cdot ky\right)} \cdot -0.16666666666666666, ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)} \cdot \sin th \]

   11. Simplified 24.3%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky, \left(ky \cdot ky\right) \cdot -0.16666666666666666, ky\right)}}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)} \cdot \sin th \]

   if 5.0000000000000001e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{\mathsf{fma}\left(ky, -0.16666666666666666 \cdot \left(ky \cdot ky\right), ky\right)}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 15.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-317}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (sin th)
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
      1e-317)
   (* -0.16666666666666666 (* th (* th th)))
   (fma
    th
    (* (* th th) (fma 0.008333333333333333 (* th th) -0.16666666666666666))
    th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(th) * (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0))))) <= 1e-317) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, ((th * th) * fma(0.008333333333333333, (th * th), -0.16666666666666666)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(th) * Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))) <= 1e-317)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(Float64(th * th) * fma(0.008333333333333333, Float64(th * th), -0.16666666666666666)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-317], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(N[(th * th), $MachinePrecision] * N[(0.008333333333333333 * N[(th * th), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 1.00000023e-317

   1. Initial program 92.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 18.3

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 18.3%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 11.5

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 11.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 18.0

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 18.0%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 1.00000023e-317 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 85.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 29.4

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 29.4%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left({th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, {th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{{th}^{2} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right)}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\left(th \cdot th\right)} \cdot \left(\frac{1}{120} \cdot {th}^{2} - \frac{1}{6}\right), th\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\left(\frac{1}{120} \cdot {th}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}, th\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \left(\frac{1}{120} \cdot {th}^{2} + \color{blue}{\frac{-1}{6}}\right), th\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{120}, {th}^{2}, \frac{-1}{6}\right)}, th\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(\frac{1}{120}, \color{blue}{th \cdot th}, \frac{-1}{6}\right), th\right) \]

      12. lower-*.f64 20.6

          \[\leadsto \mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, \color{blue}{th \cdot th}, -0.16666666666666666\right), th\right) \]

   8. Simplified 20.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin th \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-317}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, \left(th \cdot th\right) \cdot \mathsf{fma}\left(0.008333333333333333, th \cdot th, -0.16666666666666666\right), th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* (sin th) (/ ky kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = sin(th) * (ky / kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-9) then
        tmp = sin(th) * (ky / kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-9) {
		tmp = Math.sin(th) * (ky / kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 5e-9:
		tmp = math.sin(th) * (ky / kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(sin(th) * Float64(ky / kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = sin(th) * (ky / kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(N[Sin[th], $MachinePrecision] * N[(ky / kx), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{kx + \frac{1}{2} \cdot \frac{{ky}^{2}}{kx}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{1}{2} \cdot \frac{{ky}^{2}}{kx} + kx}} \cdot \sin th \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2}}{kx} \cdot \frac{1}{2}} + kx} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\frac{{ky}^{2} \cdot \frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{{ky}^{2} \cdot \frac{\frac{1}{2}}{kx}} + kx} \cdot \sin th \]

      5. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{kx} + kx} \cdot \sin th \]

      6. associate-*r/ N/A

         \[\leadsto \frac{\sin ky}{{ky}^{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{kx}\right)} + kx} \cdot \sin th \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left({ky}^{2}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{ky \cdot ky}, \frac{1}{2} \cdot \frac{1}{kx}, kx\right)} \cdot \sin th \]

      10. associate-*r/ N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{\frac{1}{2} \cdot 1}{kx}}, kx\right)} \cdot \sin th \]

      11. metadata-eval N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \frac{\color{blue}{\frac{1}{2}}}{kx}, kx\right)} \cdot \sin th \]

      12. lower-/.f64 25.0

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(ky \cdot ky, \color{blue}{\frac{0.5}{kx}}, kx\right)} \cdot \sin th \]

   8. Simplified 25.0%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(ky \cdot ky, \frac{0.5}{kx}, kx\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 24.6

          \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   11. Simplified 24.6%

       \[\leadsto \color{blue}{\frac{ky}{kx}} \cdot \sin th \]

   if 5.0000000000000001e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;\sin th \cdot \frac{ky}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \frac{th}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* ky (fma -0.16666666666666666 (/ (* th (* th th)) kx) (/ th kx)))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = ky * fma(-0.16666666666666666, ((th * (th * th)) / kx), (th / kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(ky * fma(-0.16666666666666666, Float64(Float64(th * Float64(th * th)) / kx), Float64(th / kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(ky * N[(-0.16666666666666666 * N[(N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision] + N[(th / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 35.8

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 35.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left(\frac{-1}{6} \cdot \frac{{th}^{3}}{kx} + \frac{th}{kx}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{ky \cdot \left(\frac{-1}{6} \cdot \frac{{th}^{3}}{kx} + \frac{th}{kx}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto ky \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{th}^{3}}{kx}, \frac{th}{kx}\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{th}^{3}}{kx}}, \frac{th}{kx}\right) \]

       4. cube-mult N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{th \cdot \left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       5. unpow2 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{{th}^{2}}}{kx}, \frac{th}{kx}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{th \cdot {th}^{2}}}{kx}, \frac{th}{kx}\right) \]

       7. unpow2 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{\left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{\left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       9. lower-/.f64 22.2

          \[\leadsto ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \color{blue}{\frac{th}{kx}}\right) \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \frac{th}{kx}\right)} \]

   if 5.0000000000000001e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 22.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-9}:\\ \;\;\;\;ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \frac{th}{kx}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-9)
   (* ky (fma -0.16666666666666666 (/ (* th (* th th)) kx) (/ th kx)))
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-9) {
		tmp = ky * fma(-0.16666666666666666, ((th * (th * th)) / kx), (th / kx));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 5e-9)
		tmp = Float64(ky * fma(-0.16666666666666666, Float64(Float64(th * Float64(th * th)) / kx), Float64(th / kx)));
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e-9], N[(ky * N[(-0.16666666666666666 * N[(N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision] + N[(th / kx), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-9

   1. Initial program 94.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.8

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 35.8

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 35.8%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{ky \cdot \left(\frac{-1}{6} \cdot \frac{{th}^{3}}{kx} + \frac{th}{kx}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{ky \cdot \left(\frac{-1}{6} \cdot \frac{{th}^{3}}{kx} + \frac{th}{kx}\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto ky \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \frac{{th}^{3}}{kx}, \frac{th}{kx}\right)} \]

       3. lower-/.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\frac{{th}^{3}}{kx}}, \frac{th}{kx}\right) \]

       4. cube-mult N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{th \cdot \left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       5. unpow2 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{{th}^{2}}}{kx}, \frac{th}{kx}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{\color{blue}{th \cdot {th}^{2}}}{kx}, \frac{th}{kx}\right) \]

       7. unpow2 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{\left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \frac{th \cdot \color{blue}{\left(th \cdot th\right)}}{kx}, \frac{th}{kx}\right) \]

       9. lower-/.f64 22.2

          \[\leadsto ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \color{blue}{\frac{th}{kx}}\right) \]

   11. Simplified 22.2%

       \[\leadsto \color{blue}{ky \cdot \mathsf{fma}\left(-0.16666666666666666, \frac{th \cdot \left(th \cdot th\right)}{kx}, \frac{th}{kx}\right)} \]

   if 5.0000000000000001e-9 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 61.5

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 61.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 38.7

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 38.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 22.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-15}:\\ \;\;\;\;\frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot \left(th \cdot th\right), th\right)}{kx}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-15)
   (/ (* ky (fma -0.16666666666666666 (* th (* th th)) th)) kx)
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-15) {
		tmp = (ky * fma(-0.16666666666666666, (th * (th * th)), th)) / kx;
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-15)
		tmp = Float64(Float64(ky * fma(-0.16666666666666666, Float64(th * Float64(th * th)), th)) / kx);
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-15], N[(N[(ky * N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]), $MachinePrecision] / kx), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.0000000000000001e-15

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 50.5

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 50.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)}\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \left(th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 35.4

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 35.4%

      \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + {\sin ky}^{2}}} \cdot \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(1 + \frac{1}{2} \cdot \frac{{\sin ky}^{2}}{{kx}^{2}}\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\sin ky}{kx \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{{\sin ky}^{2}}{{kx}^{2}} + 1\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       2. distribute-lft-in N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{kx \cdot \left(\frac{1}{2} \cdot \frac{{\sin ky}^{2}}{{kx}^{2}}\right) + kx \cdot 1}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       3. associate-*r* N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\left(kx \cdot \frac{1}{2}\right) \cdot \frac{{\sin ky}^{2}}{{kx}^{2}}} + kx \cdot 1} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       4. *-rgt-identity N/A

          \[\leadsto \frac{\sin ky}{\left(kx \cdot \frac{1}{2}\right) \cdot \frac{{\sin ky}^{2}}{{kx}^{2}} + \color{blue}{kx}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx \cdot \frac{1}{2}, \frac{{\sin ky}^{2}}{{kx}^{2}}, kx\right)}} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(\color{blue}{kx \cdot \frac{1}{2}}, \frac{{\sin ky}^{2}}{{kx}^{2}}, kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       7. lower-/.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx \cdot \frac{1}{2}, \color{blue}{\frac{{\sin ky}^{2}}{{kx}^{2}}}, kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx \cdot \frac{1}{2}, \frac{\color{blue}{{\sin ky}^{2}}}{{kx}^{2}}, kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx \cdot \frac{1}{2}, \frac{{\color{blue}{\sin ky}}^{2}}{{kx}^{2}}, kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       10. unpow2 N/A

           \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx \cdot \frac{1}{2}, \frac{{\sin ky}^{2}}{\color{blue}{kx \cdot kx}}, kx\right)} \cdot \mathsf{fma}\left(th, \frac{-1}{6} \cdot \left(th \cdot th\right), th\right) \]

       11. lower-*.f64 21.7

           \[\leadsto \frac{\sin ky}{\mathsf{fma}\left(kx \cdot 0.5, \frac{{\sin ky}^{2}}{\color{blue}{kx \cdot kx}}, kx\right)} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   11. Simplified 21.7%

       \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{fma}\left(kx \cdot 0.5, \frac{{\sin ky}^{2}}{kx \cdot kx}, kx\right)}} \cdot \mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right) \]

   12. Taylor expanded in ky around 0

       \[\leadsto \color{blue}{\frac{ky \cdot \left(th + \frac{-1}{6} \cdot {th}^{3}\right)}{kx}} \]
   13. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{ky \cdot \left(th + \frac{-1}{6} \cdot {th}^{3}\right)}{kx}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{ky \cdot \left(th + \frac{-1}{6} \cdot {th}^{3}\right)}}{kx} \]

       3. +-commutative N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{3} + th\right)}}{kx} \]

       4. lower-fma.f64 N/A

          \[\leadsto \frac{ky \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {th}^{3}, th\right)}}{kx} \]

       5. cube-mult N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot \left(th \cdot th\right)}, th\right)}{kx} \]

       6. unpow2 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\frac{-1}{6}, th \cdot \color{blue}{{th}^{2}}, th\right)}{kx} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{th \cdot {th}^{2}}, th\right)}{kx} \]

       8. unpow2 N/A

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(\frac{-1}{6}, th \cdot \color{blue}{\left(th \cdot th\right)}, th\right)}{kx} \]

       9. lower-*.f64 20.1

          \[\leadsto \frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot \color{blue}{\left(th \cdot th\right)}, th\right)}{kx} \]

   14. Simplified 20.1%

       \[\leadsto \color{blue}{\frac{ky \cdot \mathsf{fma}\left(-0.16666666666666666, th \cdot \left(th \cdot th\right), th\right)}{kx}} \]

   if 1.0000000000000001e-15 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 81.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 60.9

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 38.4

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 38.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 20.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 7.5 \cdot 10^{-33}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 7.5e-33)
   (* -0.16666666666666666 (* th (* th th)))
   (fma th (* -0.16666666666666666 (* th th)) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 7.5e-33) {
		tmp = -0.16666666666666666 * (th * (th * th));
	} else {
		tmp = fma(th, (-0.16666666666666666 * (th * th)), th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 7.5e-33)
		tmp = Float64(-0.16666666666666666 * Float64(th * Float64(th * th)));
	else
		tmp = fma(th, Float64(-0.16666666666666666 * Float64(th * th)), th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 7.5e-33], N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(th * N[(-0.16666666666666666 * N[(th * th), $MachinePrecision]), $MachinePrecision] + th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 7.5000000000000001e-33

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 3.6

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 3.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 3.5

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 3.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

   9. Taylor expanded in th around inf

      \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

       2. cube-mult N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

       5. unpow2 N/A

          \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

       6. lower-*.f64 16.9

          \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

   11. Simplified 16.9%

       \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]

   if 7.5000000000000001e-33 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 82.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lower-sin.f64 59.2

         \[\leadsto \color{blue}{\sin th} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

      3. *-rgt-identity N/A

         \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

      7. lower-*.f64 37.4

         \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0255:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{{\sin kx}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0255)
   (* (sin th) (/ (sin ky) (hypot kx (sin ky))))
   (* (sqrt (/ 1.0 (pow (sin kx) 2.0))) (* (sin ky) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0255) {
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sqrt((1.0 / pow(sin(kx), 2.0))) * (sin(ky) * sin(th));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0255) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sqrt((1.0 / Math.pow(Math.sin(kx), 2.0))) * (Math.sin(ky) * Math.sin(th));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.0255:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sqrt((1.0 / math.pow(math.sin(kx), 2.0))) * (math.sin(ky) * math.sin(th))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0255)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sqrt(Float64(1.0 / (sin(kx) ^ 2.0))) * Float64(sin(ky) * sin(th)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.0255)
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sqrt((1.0 / (sin(kx) ^ 2.0))) * (sin(ky) * sin(th));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0255], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0254999999999999984

   1. Initial program 86.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 60.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 60.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 73.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 73.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   if 0.0254999999999999984 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \sin th\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right)} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      4. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin th} \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      5. lower-sin.f64 N/A

         \[\leadsto \left(\sin th \cdot \color{blue}{\sin ky}\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \]

      10. lower-sin.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \]

      11. lower-pow.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \]

      12. lower-sin.f64 99.3

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}} \]

       2. lower-pow.f64 N/A

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}}}} \]

       3. lower-sin.f64 59.8

          \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2}}} \]

   11. Simplified 59.8%

       \[\leadsto \left(\sin th \cdot \sin ky\right) \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.0255:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{{\sin kx}^{2}}} \cdot \left(\sin ky \cdot \sin th\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.0255:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.0255)
   (* (sin th) (/ (sin ky) (hypot kx (sin ky))))
   (* (sin th) (* (sin ky) (sqrt (/ 1.0 (pow (sin kx) 2.0)))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0255) {
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	} else {
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / pow(sin(kx), 2.0))));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.0255) {
		tmp = Math.sin(th) * (Math.sin(ky) / Math.hypot(kx, Math.sin(ky)));
	} else {
		tmp = Math.sin(th) * (Math.sin(ky) * Math.sqrt((1.0 / Math.pow(Math.sin(kx), 2.0))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.0255:
		tmp = math.sin(th) * (math.sin(ky) / math.hypot(kx, math.sin(ky)))
	else:
		tmp = math.sin(th) * (math.sin(ky) * math.sqrt((1.0 / math.pow(math.sin(kx), 2.0))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.0255)
		tmp = Float64(sin(th) * Float64(sin(ky) / hypot(kx, sin(ky))));
	else
		tmp = Float64(sin(th) * Float64(sin(ky) * sqrt(Float64(1.0 / (sin(kx) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.0255)
		tmp = sin(th) * (sin(ky) / hypot(kx, sin(ky)));
	else
		tmp = sin(th) * (sin(ky) * sqrt((1.0 / (sin(kx) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.0255], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] * N[(N[Sin[ky], $MachinePrecision] * N[Sqrt[N[(1.0 / N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.0254999999999999984

   1. Initial program 86.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. lower-*.f64 60.4

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Simplified 60.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{kx \cdot kx} + {\sin ky}^{2}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{kx \cdot kx + \color{blue}{\sin ky \cdot \sin ky}}} \cdot \sin th \]

      3. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

      4. lower-sin.f64 73.6

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(kx, \color{blue}{\sin ky}\right)} \cdot \sin th \]

   8. Simplified 73.6%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(kx, \sin ky\right)}} \cdot \sin th \]

   if 0.0254999999999999984 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      2. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      3. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      4. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      5. lower-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      6. lower-sin.f64 99.5

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   5. Simplified 99.5%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

   6. Taylor expanded in ky around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

      2. lower-sin.f64 N/A

         \[\leadsto \left(\color{blue}{\sin ky} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\sin ky \cdot \color{blue}{\sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      5. lower-+.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right) \cdot \sin th \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      7. lower-sin.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}}\right) \cdot \sin th \]

      8. lower-pow.f64 N/A

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}}\right) \cdot \sin th \]

      9. lower-sin.f64 99.3

         \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}}\right) \cdot \sin th \]

   8. Simplified 99.3%

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       2. lower-pow.f64 N/A

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{\color{blue}{{\sin kx}^{2}}}}\right) \cdot \sin th \]

       3. lower-sin.f64 59.9

          \[\leadsto \left(\sin ky \cdot \sqrt{\frac{1}{{\color{blue}{\sin kx}}^{2}}}\right) \cdot \sin th \]

   11. Simplified 59.9%

       \[\leadsto \left(\sin ky \cdot \sqrt{\color{blue}{\frac{1}{{\sin kx}^{2}}}}\right) \cdot \sin th \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;kx \leq 0.0255:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\mathsf{hypot}\left(kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 10.9% accurate, 39.5× speedup

Math

\[\begin{array}{l} \\ -0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* -0.16666666666666666 (* th (* th th))))

C

double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Fortran

real(8) function code(kx, ky, th)
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (-0.16666666666666666d0) * (th * (th * th))
end function

Java

public static double code(double kx, double ky, double th) {
	return -0.16666666666666666 * (th * (th * th));
}

Python

def code(kx, ky, th):
	return -0.16666666666666666 * (th * (th * th))

Julia

function code(kx, ky, th)
	return Float64(-0.16666666666666666 * Float64(th * Float64(th * th)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = -0.16666666666666666 * (th * (th * th));
end

Wolfram

code[kx_, ky_, th_] := N[(-0.16666666666666666 * N[(th * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lower-sin.f64 22.7

      \[\leadsto \color{blue}{\sin th} \]

5. Simplified 22.7%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto \color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto th \cdot \color{blue}{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + th \cdot 1} \]

   3. *-rgt-identity N/A

      \[\leadsto th \cdot \left(\frac{-1}{6} \cdot {th}^{2}\right) + \color{blue}{th} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(th, \frac{-1}{6} \cdot {th}^{2}, th\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(th, \color{blue}{\frac{-1}{6} \cdot {th}^{2}}, th\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(th, \frac{-1}{6} \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

   7. lower-*.f64 15.2

      \[\leadsto \mathsf{fma}\left(th, -0.16666666666666666 \cdot \color{blue}{\left(th \cdot th\right)}, th\right) \]

8. Simplified 15.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(th, -0.16666666666666666 \cdot \left(th \cdot th\right), th\right)} \]

9. Taylor expanded in th around inf

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]
10. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{-1}{6} \cdot {th}^{3}} \]

    2. cube-mult N/A

       \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot \left(th \cdot th\right)\right)} \]

    3. unpow2 N/A

       \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{{th}^{2}}\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{-1}{6} \cdot \color{blue}{\left(th \cdot {th}^{2}\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{-1}{6} \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

    6. lower-*.f64 12.9

       \[\leadsto -0.16666666666666666 \cdot \left(th \cdot \color{blue}{\left(th \cdot th\right)}\right) \]

11. Simplified 12.9%

    \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(th \cdot \left(th \cdot th\right)\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))